1. THE DIXIT-STIGLITZ MODEL
Lecture 5
SPATIAL ECONOMY:
THE DIXIT-STIGLITZ MODEL
By Carlos Llano,
References for the slides:
Fujita, Krugman and Venables: Spatial Economy. Ariel Economía, 1999.
Brakman S., Garretsen H. van Marrewijk C. (2009): The New Introduction to Geographical Economics. Cambridge University Press.
The World Bank (2008): “Reshaping economic geography”. WB report.

2. Outline Introduction The Dixit-Stiglitz: Descriptions and assumptions.
Demand.
Transport costs and multiple locations.
Conclusion

3. 1. Introduction: Geography-Economy in 3D
TWB (2009): Reshaping Economic Geography.
Density
Distance
Division
BGM (2010): The New introduction to Geographical Economics. Lesson 1:
Economic Agglomeration
Economic Interaction.

4. 1. Introduction: density = spatial agglomeration

5. 1. Introduction 5

6. 1. Introduction. Density: I-Current situation
Country level: the concentration of economic activity also occurs within countries, and increases with the “income level”. This agglomeration is not an artifact of the spatial unit used.
Country
GDPpc
# of administ. Areas
Concentration By
Tanzania
324 21 15
%GDP in the leading area
Administrative
Italy
19,480
areas
France
22,548 22 29
Sweden
31,197
By Statistical
Tajikistan
204 5
30.2
% household consumption in the leading area
Mongolia
406
34.6
El Salvador
1,993
43.9
Brazil
3,597
51.6
Argentina
7,488
64.7
By Land
Ghana
211
227,540
0.48
Spatial Gini coefficient
Lao
231
230,800
Poland
3,099
311,888
0.52
New Zealand
11,552
267,990
0.55
Norway
27,301
304,280
0.64

7. 1. Introduction. Density: II-Dynamics
Country level: Spatial inequality of regions within countries rose and remained high before slowly declining, following an inverted-U relationship. WB, pp86.
II-Dynamics

8. 1. Introduction. Density: II-Dynamics
Country level:
Economic development, in its early stages, is accompanied by a rapidly rising spatial concentration in a country.
Leading areas benefit most from this compression and growth. TWB, pp86.

9. 2. The Dixit-Stiglitz Model
The Dixit-Stiglitz model is the most used framework for describing a monopolistic competition models (Dixit-Stiglitz, 1977).
Since the 70’s, its use in the field of international trade has been fundamental. It is the starting point for the New Economic Geography (NEG): agglomeration, economies of scale, transportation cost.
Fujita, Krugman and Venables (1999) present a spatial version of the DSM:
2 regions; 1 mobile production factor (L= labor).
2 products:
Agriculture: residual sector, perfect competitive, constant returns to scale; no transportation costs.
Manufacturing: differentiated goods (n varieties); scale economies; monopolistic competition; transportation costs. 9

10. 2. The Dixit-Stiglitz Model
Structure of the Dixit-Stiglitz spatial model:
Solution to the consumer’s problem
Multiple Locations and Transportation Costs
Producer Behavior
The Price Index Effect and the Home Market Effect
Equilibrium 10

11. 2. The Dixit-Stiglitz Model
Consumer Behavior: Utility function
Every consumer shares the same Cobb-Douglas tastes for the two type of goods (M, A).
M= composite index of the manufactured goods.
A= consumption of the agricultural good.
Mu (μ): constant: expenditure share in manufactured goods.
M is a sub-utility function defined over a continuum of varieties of manufactures:
m(i): consumption of each available variety (i),
n: range of varieties.
M is defined by a constant-elasticity-of-substitution (CES):
Rho (ρ): intensity of the preference for variety (love for variety)
If ρ=1, differentiated goods are nearly perfect substitutes (low love for variety)
If ρ=0, the desire to consume a greater variety of manufactured goods increases. 11

12. 2. The Dixit-Stiglitz Model
Consumers Behavior:
We define sigma (σ) as:
σ: elasticity of substitution between any 2 varieties
The consumer’s problem: maximize utility defined by the function U subject to the budget constraint.
We solve it in 2 steps:
First, the consumption of varieties will be optimized:
The ideal consumption of each variety will be given by the combination that ensures utility with the minimum cost (given the relative prices of each variety).
Once the consumption of varieties in generic terms has been optimized (for every M), then, the desired quantity of A and M will be chosen according to the relative prices of both goods. 12

13. 2. The Dixit-Stiglitz Model
The Consumers Behavior: the budget constraint
PA= Price of the agricultural goods.
A= consumption of the agricultural good.
p(i)= price of each variety (i) of manufacturing product.
m(i)= quantity of each variety (i).
To maximize the utility U subject to the budget constraint Y, there are 2 steps:
Whatever the value of the manufacturing composite (M), each m(i) needs to be chosen so as to minimize the cost of attaining de M (Phase I).
Afterwards, the step is to distribute the total income (Y) between agriculture (A) and manufactures (M) in aggregate (Phase II). 13

14. 2. The Dixit-Stiglitz Model
1. Consumer Behavior: Phase I:
1. Minimize expenditure for any given M :
PA= Price of the agricultural goods.
A= consumption of the agricultural good.
p(i)= price of each variety (i) of manufacturing product.
m(i)= quantity of each variety (i). 14

15. 2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase II:
Now we have to divide the total income (Y) between the two goods, M and A. We will do it by maximizing U constrained to the optimal expenditure derived from minimizing the cost of attaining M.
PA= Price of the agricultural goods.
G= Manufactures’ Price Index
A= consumption of the agricultural good.
p(i)= price of each variety (i) of manufacturing product.
m(i)= quantity of each variety (i). 15

16. 2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I + Phase II:
Now FKV introduce a variation of the DS Model:
They make that the range of manufactures on offer becomes an endogenous variable.
If ↑n → ↓G (manufactures’ price index), because consumers value variety.
Therefore ↓ Cost of attaining a given level of utility.
To prove it, we assume that all manufactures are available at the same price, pM . Then, the price index G becomes: 16

17. 2. The Dixit-Stiglitz Model
1. Consumer behavior: Phase I + Phase II:
The relationship between G and n depends on the elasticity of substitution between varieties σ
The lower is σ (the more differentiated are varieties) → the greater is the reduction in G caused by an increase in the number of varieties.
Changing the range of products available also shifts demand curves for existing varieties.
To prove it, we look at the demand curve for a single variety:
When Δn → ↓G , the demand m(j) shifts downward,
Important: it allows us to know the equilibrium n:
If Δn → Δ competition → shifts downward the existing products m(j) and reduces the sales of those varieties (evolution to more firms with profit=0) 17

18. 2. The Dixit-Stiglitz Model
We consider the existence of R possible discrete locations.
Each variety is produced in only one location and all varieties produced in a particular location are symmetric in technology and price.
nr= number of varieties in location r.
pmr= FOB price of manufacturing in location r.
Agricultural and Manufactured products can be shipped between locations incurring in transport costs:
Iceberg costs: if a unit of a good is shipped from a location r to another location s, only a fraction of the original unit actually arrives. The rest is “lost” (melted) in the trip (transport cost).
The constant represents the amount of the agricultural good dispatched per unit received in s.
2. Multiple locations and transportation cost: iceberg costs 18

19. 2. The Dixit-Stiglitz Model
2. Multiple locations and transportation cost: CIF prices
If pmr is the FOB price of the manufacturing product in location r, and there are iceberg transport costs, the CIF price when delivered to location s is given by:
Then, the manufacturing price index (Gs) may take a different value in each location according to the location s where it is consumed:
Price index in s of manufactures produced in r
Consumption demand in location s for a product produced in r
Ys= income for location s: this gives the consumption of the variety in s. 19

20. 2. The Dixit-Stiglitz Model
2. Multiple locations and transportation cost: CIF prices
As a consequence, summing across locations in which the product is sold, the total sales of a single location r variety is:
I have to produce Tmrs in r, knowing that a portion 1/ Tmrs is lost during the trip (transportation cost)
Important consequences:
Sales depend on: income and the price index in each location, on the transportation costs and the mill price.
Because the delivered prices of the same variety at all consumption locations change proportionally to the mill price, and because each consumer’s demand for a variety has a constant price elasticity sigma (σ), the elasticity of the aggregate demand for each variety with respect to its mill price is also sigma (σ), regardless of the spatial distribution of consumers. 20

21. 2. The Dixit-Stiglitz Model
3. Producer Behavior:
The agricultural goods is produced with constant returns;
Manufacturing involves economies of scale at the level of the variety (internal).
Technology is the same for all varieties and in all locations:
The only input is labor L, the production of a quantity qM of any variety at any given location requires labor input lM , given by:
With increasing returns to scale, consumer’s preference for variety, and the unlimited number of potential varieties of manufactured goods, no firm will choose to produce the same variety supplied by another firm,
Each variety is produced in only one location by a single specialized firm,
The number of manufacturing firms is the same as the number of available varieties. 21

22. 2. The Dixit-Stiglitz Model
3. Producer Behavior: Profit maximization
Firms maximize profits with a given income (sales) and with given costs (according to the wages)
Revenues (sales)
Costs: F+V (given the wages wr) 22

23. 2. The Dixit-Stiglitz Model
3. Producer Behavior: wages
If firms in “r” make profit =0, they produce q*. With this q* they satisfy the demand of products from “r”:
We can turn this equation around and say that active firms break even if and only if the price they charge satisfies:
Using the price rule (*) we get:
(*)
This is the wage equation: it gives the manufacturing wage at which firms in each location break even, given the income levels and price indices in all locations and the costs of shipping into these locations:
The wage increases with the income (Ys) at location s, the access to location s from location r (Tmrs), and the less competition the firm faces in location s (G decreases with n) 23

24. 2. The Dixit-Stiglitz Model
3. Producer Behavior: wages
Real wages: real income at each location is proportional to nominal income deflated by the cost-of-living index,
This means that the real wage of manufacturing workers in location r, denoted by ωrM is

25. 2. The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
We consider an economy with 2 regions, that produce 2 manufacturing varieties:
These pairs of equations are symmetric, and so its’ solutions.
So, if L1=L2; Y1=Y2, then there is a solution with G1=G2 and with w1=w2.
We can explore the relationships contained in the price indices and wage equations by linearizing them around the symmetric equilibrium:
An increase in a variable in R1 is associated with a decrease in R2 but of equal absolute magnitude.
So letting dG=dG1=-dG2, and so on, we derive, by differentiating the price indices and wage equations respectively, and we get:

26. 2. The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
Direct effect of the variation in the location of firms on the Price Index (G).
How relative demand affects the location of M.
[Eq 1]: Price Index Effect: We suppose that the supply of labor is perfectly elastic, so that dw=0. Bearing in mind that 1-σ <0 and that T>1, the equation implies that a change dL/L in manufacturing employment has a negative effect on the price index, dG/G.
Conclusion: the location with a larger manufacturing sector also has a lower price index for manufactured goods, simply because a smaller proportion of this region’s manufacturing consumption bears transport costs.

27. 2. The Dixit-Stiglitz Model
5. The price index effect and the Home Market Effect
Now , let us consider how relative demand affects the location of manufacturing. It is convenient to define a new variable, Z,
Z is an index of trade cost, with value between 0-1:
Z=0, if trade is costless (T=1);
Z=1, if trade is impossible (T=0).
Using the definition of Z and eliminating dG/G, we have
If dw=0, supply of labor is perf. elastic: Home market effect: A 1% change in demand for manufactures (dY/Y) causes a 1/Z % (>1) change in the employment, and the production of manufactures (dL/L).
The location with the larger home market has a more than proportional larger manufacturing sector (industrial agglomeration) and therefore also tends to export manufactured goods.
If dw>0, positive supply of labor : part of the home market advantages goes to higher wages instead of exports
Locations with a larger home market (demand) tends to offer a higher nominal wage (qualified labor agglomeration). 27

28. 2. The Dixit-Stiglitz Model
6. The “No-Black-Hole” Condition
We are not interested in economies in which increasing returns are so strong that the forces working toward agglomeration always prevail, and the economy tends to collapse into a point. (Everyone to NY).
To avoid this “black-hole location” effect, we usually impose what we call the “no black hole condition”:

29. In search of the Home Market Effect (HME)
In classic Trade theory models, where industries are suppose to have CRS, a larger demand of a product in a country will imply being a net importer of such product.
In NTT, due to the Home Market Effect (HME):
Industries with Increasing Returns to Scale (IRS), and larger demand in a country/region will imply more than proportional production, and to be a net-exporter of that product.
David & Wenstein test empirically the existence of HME:
DW 1996, 1997, 2003: 22 OECD countries; 26 industries.

30. In search of the Home Market Effect (HME)
Journal of International Economics 59 (2003) 1–23

31. In search of the Home Market Effect (HME)
n= # industries
g= # goods
c= # countries
ROW= rest of the world
Xngc= output of product g in industry n in country c.
Dngc= demand of product g in industry n in country c.

32. In search of the Home Market Effect (HME)
Xngc= output of product g in industry n in country c.
Ω= technology matrix
V= factor endowments of country c

33. In search of the Home Market Effect (HME)

34. The spatial wage structure and real market potential
In neoclassical trade theory, there is no foundation for spatial wage structure.
In fact, H-O predicts a complete factor price equalization due to international trade.
In New Trade Theory –without transport costs-, although in autarky the larger market will pay higher wages, by trade, factor price equalization is also predicted.
The basic NEG model (Krugman, 1991) predicts a spatial wage structure in favor of the “core”.
In other versions of the CP model, different stages are considered, with increasing-stable-decreasing differences in terms of “real wages” along with the reduction of transport costs.

35. The spatial wage structure and real market potential
Evidence for México (Hanson, 1997):
He analyzed the evolution of wage structure before and after the shift in trade policy: from protectionism to NAFTA, testing 2 hypothesis:
H1: the region´s wage relative to Mexico city are lower when transport cost (distance) are higher.
H2: Trade liberalization led to a compression of regional wage differentials.

36. He found empirical evidence for Hyp1, but not for Hyp2. Why
He found empirical evidence for Hyp1, but not for Hyp2. Why? Trade liberalization shifted the weight center of the economy north-wards, reducing the centrality of México-City.
Figure 5.5 Map of Mexico
Source: BGM-2010: